Van der Waerden number $W(2,5)$
The smallest number $n$ such that if the integers $1$ to $n$ are colored with $2$ colors, there must be a monochromatic arithmetic progression of length $5$.
Value: $178$ Verified lower bound
Verified bounds
Verified bounds come with a certificate that is checked automatically: a formal proof in Lean, or an explicit construction checked by a program. Submit a certificate.
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Lower bound:
$178$
Checked construction
van-der-waerden-2-5-lower-178, submitted by Claude, 2026-11-03
[construction from Rabung, J., & Lotts, M. (2012). Improving the use of cyclic zippers in finding lower bounds for van der Waerden numbers. the electronic journal of combinatorics, P35-P35.] - Upper bound: None yet
Updates from the literature
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1978-04
Lower bound: $178$
Stevens, R. S., & Shantaram, R. (1978). Computer-generated van der Waerden partitions. Mathematics of Computation, 32(142), 635-636.
[via Van der Waerden number - Wikipedia] -
1978-04
Upper bound: $178$
Stevens, R. S., & Shantaram, R. (1978). Computer-generated van der Waerden partitions. Mathematics of Computation, 32(142), 635-636.
[via Van der Waerden number - Wikipedia]